7
$\begingroup$

A rectangle with side length a and b are in ratio $a : b = (n+1)^2 : n^2$, where n is a positive integer.

Is it possible to cut each such rectangle into two pieces, which can be put together to build a square?

$\endgroup$
16
$\begingroup$

The answer to your question:

Yes, this is always possible

Proof:

This solution, shown with $n = 6$, can be applied for all $n$. The rectangle with dimensions $n^2$ by $(n+1)^2$ is divided into $n\times(n+1)$ congruent rectangles with dimensions $n$ by $n+1$ such that there are $n$ rows of height $n$ and $n+1$ columns of width $n+1$. The rectangle is cut into two congruent pieces along a stair-step diagonal. These pieces can be arranged to form a square with edge length $n\times(n+1)$, divided into $n+1$ rows of height $n$ and $n$ columns of width $n+1$. Solution

$\endgroup$
7
  • 4
    $\begingroup$ The cut is very hard to make out in your picture. Could you color it stronger? And maybe add a text description of what you are doing for visually impaired users ? $\endgroup$
    – Falco
    Dec 3 '21 at 10:32
  • 1
    $\begingroup$ Your rectangle seems to have side in ratio n:(n=1) - 6 and 7 specifically. The case for n=6 would have sides of lengths 49 and 36, which is not the same shape, and does not enable a simplistic 1x1 "staircase" cut. $\endgroup$
    – AdamV
    Dec 3 '21 at 12:44
  • 3
    $\begingroup$ @AdamV if you read the (less than 20 words of) explanation given, you'll notice that the non-square looking small rectangles (that go into the square 6 times horizontally and 7 times vertically) do not actually represent unit squares. $\endgroup$
    – Bass
    Dec 3 '21 at 14:44
  • $\begingroup$ @Bass it's not less than 20 words ;-P $\endgroup$
    – loopy walt
    Dec 3 '21 at 19:57
  • 3
    $\begingroup$ @Falco Your suggestions have been applied. $\endgroup$ Dec 3 '21 at 22:48
7
$\begingroup$

I managed to do it like this:

Start with a rectangle of size 16 x 9 and cut into two parts as shown below:

1
Then place the pieces together like this to form a 12 x 12 square:

2

$\endgroup$
3
  • 2
    $\begingroup$ The question is asking if this is possible for all n. $\endgroup$ Dec 2 '21 at 23:51
  • $\begingroup$ @DanielMathias OK I see the "each" in the question now. Looks like you supplied the correct answer for that. Now I wonder if other rectangles can be cut/pasted into squares or only these kinds? $\endgroup$
    – JS1
    Dec 2 '21 at 23:57
  • $\begingroup$ The steps of this staircase are each n x (n+1). That will always work because there will always be n divisions of n into n^2, and n+1 divisions of n+1 into (n+1)^2, enabling the "shift and lift" to equalise them so you end up with sides of (n+1) x n and n x (n=1) which are of course the same. $\endgroup$
    – AdamV
    Dec 3 '21 at 12:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.